Spanning Star Forest Research Articles

On the star forest polytope for trees and cycles

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

Complexity and approximability of extended Spanning Star Forest problems in general and complete graphs

A solution extension problem consists in an instance and a partial feasible solution which is given in advance and the goal is to extend this partial solution to a feasible one. Many well-known problems like Coloring Extension Problems, Traveling Salesman Problem and Routing problems, have been studied in this framework. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP-hardness, parameterized complexity, positive and negative approximation results.

The maximum weight spanning star forest problem on cactus graphs

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946–962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus.

APPROXIMATING THE SPANNING k-TREE FOREST PROBLEM

The spanning star forest problem is an interesting algorithmic problem in combinatorial optimization and finds different applications. We generalize it into the spanning k-tree forest problem, which is to find a maximum spanning forest in which each tree component has a central vertex and other vertices in the component have distance at most k away from the central vertex. We show that this new problem can be approximated with ratio [Formula: see text] in polynomial time for both undirected and directed graphs. In the weighted distance model, a ½-approximation algorithm is presented for it.

Improved approximation for spanning star forest in dense graphs

A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c?(0,1), a graph of n vertices is called c-dense if it contains at least cn 2/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ?>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c?(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.

Improved Approximation Algorithms for the Spanning Star Forest Problem

A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set.We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946–962, 2008). We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge-weighted and node-weighted) versions of the problem.Our algorithms use a non-linear rounding scheme, which might be of independent interest.

Approximating the Spanning Star Forest Problem and Its Application to Genomic Sequence Alignment

This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomial-time approximation scheme for planar graphs; (2) there is a polynomial-time $\frac{3}{5}$-approximation algorithm for graphs; (3) it is NP-hard to approximate the problem within ratio $\frac{259}{260} + \epsilon$ for graphs; (4) there is a linear-time algorithm to compute the maximum star forest of a weighted tree; (5) there is a polynomial-time $\frac{1}{2}$-approximation algorithm for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region.

Edge Colouring of K 2n with Spanning Star-Forests Receiving Distinct Colours

Let Sni be a star of size ni and let S=Sn1∪…∪Snk≠S2n−3∪S1 or S2∪S2 be a spanning star-forest of the complete graph K2n. We prove that K2n has a proper (2n−1)-edge-colouring such that all the edges of S receive distinct colours. This result is very useful in the study of total-colourings of graphs.